The complexity of multiphasic flows basically lies in the fact that the time evolution of interfaces, whose position is an unknown of the problem, may lead to their deformation, their break-up or coalescence. Moreover, interfaces obey to physical phenomena where capillary effects play an important role.
The various domains of application, where multiphasic flows are involved, are generally complex; the experimentation and measurements are quite difficult, onerous, and most often not very accurate. For instance, in nuclear safety [1], the understanding of interaction between molten corium (lava-like molten mixture of portions of nuclear reactor core) and concrete (last confinement barrier) is a major issue. An approach using direct numerical simulations allows to access to instantaneous quantities at each point of the flows.
Due to their ability to capture interfaces implicitly, diffuse interfaces models are attractive for the numerical simulations of multiphase flows. In this article, we consider a model that couples the Cahn–Hilliard system and the Navier–Stokes equation.
A. The Cahn–Hilliard Model
In diffuse interfaces theory, the interfaces are assumed to have a nonzero thickness ε (which is here a constant parameter of the model). Interfaces are considered as mixing areas, and the phase i can be represented by a smooth phase indicator c_{i} called “order parameter” (which may be understood here as the volumic fraction of the phase i). Thus, the system contains as many unknowns c_{i} as phases. These unknowns vary between 0 and 1 (values that correspond to pure phases by convention) and are linked by the relationship .
A complete derivation of this kind of models for diphasic flows is presented in Refs. [2–5]. Different extensions have been proposed for the simulations of three-phase flows in Refs. [6–8]. We consider in this article the triphasic Cahn–Hilliard model taken from Ref. [6]:
(1.1)
where M_{0}(c) is a diffusion coefficient which is called “mobility” and may depend on c = (c_{1}, c_{2}, c_{3}). The functions f are defined by:
where Σ_{T} is given by . This system is a gradient flow for the following energy functional under the constraint of volume conservation:
(1.2)
where Ω denotes an open, bounded, connected, and smooth domain of (d = 2 or 3). The “intermediate” unknowns μ_{i}, called “chemical potentials,” are the functional derivatives of the triphasic Cahn–Hilliard energy (1.2). The rather intricate expression of f let us ensure the constraint:
We introduce the hyperplane of , to simplify notation in the sequel.
The expressions of the triphasic Cahn–Hilliard potential F and the constant triplet Σ = (Σ_{1}, Σ_{2}, Σ_{3}) was derived in Ref. [6], so that the model can correctly take into account the surface tensions values σ_{12}, σ_{13}, and σ_{23} prescribed between the different pairs of phases, and it is consistent with the two-phase situations: the triphasic model has to exactly reproduce diphasic situations when one of the three phases is not present. The coefficient Σ_{i} is given as a function of the surface tensions:
(1.3)
and the triphasic potential F has the following form:
where Λ is an arbitrary smooth function of c.
Note that, in the sequel, we do not assume that the coefficients Σ_{i} are non-negative, so that the model can handle some total spreading situations. However, as it is proved in Ref. [6], the following condition is necessary to ensure the well posedness of the system:
(1.4)
This condition is equivalent to the coercivity of capillary terms and ensures that these terms bring a positive contribution to the free energy . This is detailed in the following proposition:
Proposition 1.1 ([6, Prop 2.1]). Let . There existsΣ > 0 such that, for alln ≥ 1, for allsuch that ξ_{1} + ξ_{2} + ξ_{3} = 0,
if and only if the two following conditions are satisfied:
(1.5)
This proposition will be useful in the sequel.
Remark 1. Owing to (1.3), the second part of condition (1.5) is always satisfied and consequently it is sufficient to assume that the condition (1.4) holds, for applying Proposition 1.1.
The existence of weak solutions for problem (1.1) together with initial and Neumann boundary conditions (for order parameters c_{i} and chemical potentials μ_{i}) was proved in Ref. [6] (see Ref. [9] for an alternative proof based on numerical schemes) in 2D and 3D under the following general assumptions:
The mobility M_{0} is a bounded function of class, and there exists three positive constants M_{1}, M_{2}, and M_{3} such that:
(1.6)
The Cahn–Hilliard potential F is a positive function of class that satisfies the following assumptions of polynomial growth: there exist a constant B_{1} > 0 and a real p such that 2 ≤ p < +∞ for d = 2 or 2 ≤ p ≤ 6 for d = 3, and
(1.7)
In the previous items, the notation D stands for the differentiation operator.
B. Coupling with Hydrodynamic
The coupling between the Cahn–Hilliard and Navier–Stokes systems is obtained by:
1adding a transport term u · ∇c_{i} in the evolution equation of each order parameter c_{i}, (i ∈ {1,2,3}), which is the first equation of system (1.1).
2defining the density and viscosity as smooth functions of order parameters c.
3adding a capillary forces term in the right-hand side of the momentum balance (in the Navier–Stokes equation).
Furthermore, we adopt a particular form of the Navier–Stokes equation, initially proposed in Ref. [10] (see also Refs. [5, 11]), which ensures an energy balance without using the equation of mass conservation. It relies on the following inequality:
the domain Ω_{t} being an arbitrary bounded smooth domain moving at the fluid velocity u [12].
Hence, the triphasic Cahn–Hilliard/Navier–Stokes, we study here, is constituted with following equations:
(1.8)
where the vector g stands for the gravity; the tensor Du stands for the symmetric gradient of the velocity u; the scalar p stands for the pressure; the density and viscosity are defined by:
where ϱ_{1} (resp. ϱ_{2}, ϱ_{3}) and η_{1} (resp. η_{2}, η_{3}) are the prescribed values (assumed to be constant) in phase 1 (resp. 2, resp. 3) and the function h_{λ} (λ = 0.5) is defined by:
We supplement this system with Neumann boundary conditions for order parameters c_{i} and chemical potentials μ_{i}, that is, for i = 1,2,3,
(1.9)
and with a homogeneous Dirichlet boundary condition for the velocity, that is,
(1.10)
Owing to these boundary conditions (1.9) and (1.10), we introduce the following function spaces:
Finally, we assume that the following initial condition holds:
(1.11)
where and are given.
C. Outline of the Article
In Section II, we describe the time and space discretization of the problem (1.8). We then prove, in Section III, the unconditional stability of the scheme and the existence of approximate solutions. Section IV is devoted to numerical experiments. In the last section (Section V), we prove the convergence of approximate solutions toward weak solutions of the Cahn–Hilliard/Navier–Stokes system in the case where the three fluids have the same density. In particular, we prove the following existence theorem by passing to the limit in the numerical scheme in Section V:
Theorem 1.2 (Existence of weak solutions in the homogeneous case). Assume the coefficients (Σ_{1}, Σ_{2}, Σ_{3}) satisfy the condition (1.4), the mobility satisfy (1.6), and that the Cahn–Hilliard potential F satisfy the condition (A). Assume the densities of the three fluids are equal, that is, ϱ_{1} = ϱ_{2} = ϱ_{3} = ϱ_{0}, . Consider the problem (1.8) together with initial condition (1.11) and boundary conditions (1.9) and (1.10). Then, for allt_{f} ∈]0, +∞[, there exists a weak solution (c, μ, u, p) on [0, t_{f}[ such that
II. DISCRETIZATION OF THE CAHN–HILLIARD/NAVIER–STOKES MODEL
A. Time Discretization
Let . The time domain [0, t_{f}] is uniformly discretized with a fixed time step ; we define t_{n} = nΔt, for all n ∈ [[0; N]]. We assume that the functions and (n ∈ [[0;N − 1]]) are given and we describe the system we have to solve to compute the unknowns and at time t^{n+1}.
We first describe, in two distinct paragraphs, the schemes we use to discretize the Cahn–Hilliard and Navier–Stokes equations without considering the coupling terms. We then explain, in the next two paragraphs, the reasoning leading to the discretization of the coupling terms before writing the complete scheme in the last paragraph. For more details on the time discretizations of the triphasic Cahn–Hilliard model, the reader may refer to the article [9] (and references therein). Several articles in the literature are devoted to the study of discretizations of the Navier–Stokes equation: we refer in particular to the articles [10, 13], which deal with variable density models.
Cahn–Hilliard System.
We consider a time discretization of the Cahn–Hilliard system of the form: for i = 1,2,3,
where c = (1 − β)c + β c, β ∈ [0.5, 1], and M = M_{0}((1 − α)c^{n} +α c^{n+1}), α ∈ [0, 1]; the discretization of transport terms is postponed to “Coupling Terms” “Coupling Terms” section.
This kind of discretizations was presented and studied in Ref. [9]. This is out the scope of the article. We assume that the discretization D(c^{n}, c^{n+1}) of the term f is of the form:
where d stands for a semi-implicit discretization of ∂_{i}F. We assume that the two following basic properties hold for all i ∈{1, 2, 3}:
(2.1)
(2.2)
the notation D means here the derivative of d with respect to the second variable. The assumption (2.1) is a consistency assumption and the assumption (2.2) is the counterpart of the polynomial growth assumption (1.7) on F. Many possible choices for the discretization of the term d were presented in Ref. [9]. For instance, we consider in numerical tests of this article (Section IV) the following expression:
This scheme was built to ensure the following equality:
and consequently a discrete energy equality which is obtained by multiplying the first equation of the Cahn–Hilliard system by μ, the second one by c − c, writing the equality of left-hand sides and summing for i = 1,2,3.
Navier–Stokes System.
We now present the time discretization of the momentum balance of the Navier–Stokes system:
We separately present the discretization of the different terms (1)–(3) involved in the above equation; for each of them, we give their contribution to the energy balance obtained at the discrete level by multiplying the equation by u^{n+1} and integrating on Ω.
Term (1): Using the formal equality , we choose the following discretization of Term (1) (see Ref. [13]):
Its contribution to the energy balance is:
Term (2): The Term (2) is linearized using an explicit velocity for the transport:
Its contribution to the energy balance vanishes. Indeed, for all , we have:
In particular, when we take ν^{u} = u^{n+1}, the above term vanishes.
Term (3): We discretize the Term (3) with an implicit scheme: −div (2η^{n+1}Du^{n+1}). Its contribution to the energy balance is .
Thus, we adopt the following discretization of the Navier–Stokes equation:
The discretization of the capillary forces term is described in the next paragraph.
Coupling Terms.
We give in this paragraph the discretization of coupling terms. That is the transport terms u Δ∇c_{i} in the Cahn–Hilliard equations, and the capillary forces term in the momentum balance (Navier–Stokes equation). At the continuous level, when writing the energy balance, the contributions of these two terms counterbalance each other. At the discrete level, we saw that the energy balance is obtained, for the Cahn–Hilliard system, by multiplying the transport terms by μ before summing up for i = 1,2,3 and, for the Navier–Stokes equation, by multiplying the capillary forces term by u^{n+1}.
Consequently, it is easy to see that when all the terms mentioned above are discretized with an implicit scheme (cf Ref. [14] for the diphasic case), that is, u^{n+1} · ∇c and , the balance is also true at the discrete level. However, this discretization introduces a strong coupling between the Cahn–Hilliard and Navier–Stokes systems. The discrete system is difficult to solve in practice.
It is possible to uncouple the two systems (cf Ref. [15] for the diphasic case, Ref. [11] for the triphasic case) using an explicit velocity (i.e., the velocity at time t^{n}) in the Cahn–Hilliard equation: u^{n} · ∇c. However, the contributions of the transport terms in the Cahn–Hilliard system and the contribution of the capillary forces in the Navier–Stokes equation do not counterbalance when writing the discrete energy balance, which contains the additional term: . It is difficult to attribute a sign to this term, and the scheme stability is obtained only conditionally (cf Ref. [15], assuming for instance that the ratio between the time step and the mesh size is bounded).
We first observe that it is possible to uncouple the resolution of the Navier–Stokes system and the taking into account of capillary forces. The taking into account of the capillary forces is performed during a first step which provides an intermediate velocity u* which is then used in the Cahn–Hilliard system. The Navier–Stokes system is then solved in a second step. The scheme reads:
i.Taking into account of capillary forces:
ii.Cahn–Hilliard system:
iii.Navier–Stokes system:
This discretization is unconditionally stable but the system of step (i) (Darcy problem) is still coupled with the Cahn–Hilliard equations (system (ii)).
We propose to forget for a moment the divergence free constraint imposed to u* (and consequently the associated pressure term ∇p*) in the system of the step (i). This leads to define u* as follows:
This definition of u* is explicit, and u* can be replaced by its expression in the Cahn–Hilliard system thus eliminating the coupling with the Navier–Stokes equation.
The problem is that u* is not divergence free. Nevertheless, note that the divergence of u* is of order O(Δt) and that the property u* · n = 0 is still satisfied on Γ. Now, the question is: is it possible to discretize the transport term in the Cahn–Hilliard equation to preserve its fundamental properties (volume conservation and the fact that the sum of the three order parameters remains equal to 1)? An answer is given in the next paragraph.
Transport Term in the Cahn–Hilliard System When the Velocity Is Not Divergence Free.
In this paragraph, we are interested in the form of the transport term in the Cahn–Hilliard equation when the advection velocity, denoted by u* is not divergence free but satisfy the boundary condition u*·n = 0 on Γ.
Preserving properties of the Cahn–Hilliard when the advective velocity is not divergence free may be useful in other contexts. For instance, when using an incremental projection method (cf. Ref. [16, 17]), the end step velocity is not divergence free.
The transport term may be written in the conservative or non conservative form (these two forms are not equivalent since a priori div (u*) ≠ 0):
nonconservative form: u* · ∇c_{i},
conservative form: div (c_{i}u*).
The conservative form ensures the total volume conservation of each phase (as u*·n = 0 on Γ). This is not the case for the nonconservative form as a priori . Conversely, when using the conservative form, a necessary condition to ensure that the sum of the three-order parameters c_{i} remains constant equal to 1, is div (u*) = 0. Neither the conservative form nor the nonconservative form ensures both volume conservation and the fact that the sum of the three-order parameters remains equal to 1.
We propose to use the following formulation:
where α_{i} is a constant. This formulation allows to ensure the two desired properties if . To guarantee the consistency with diphasic model, the constant α_{i} may be zero when the phase i is not present. In the sequel, we propose to choose:
This formulation allows to use an advective velocity which is not divergence free. The term −α_{i}div (u*) is added in the Cahn–Hilliard system, its role is to re-equilibrate the values of each order parameters to ensure the fact that their sum remains equal to 1. We prove in Section V that this term is of order O(h + Δt), so it does not disturb the consistency of the scheme.
Owing to this formulation of the transport term, it seems natural to adopt the following definition for the capillary forces term in the Navier–Stokes equation:
This is equivalent to modify the definition of the pressure by subtracting the term .
Time Discretization of the Cahn–Hilliard/Navier–Stokes System.
Finally, the different considerations presented in the previous paragraphs lead to propose the following scheme:
Problem 1.
Step 1: resolution of the Cahn–Hilliard system
Findsuch that, fori = 1−3,
with α_{j} a constant: .
Step 2: resolution of the Navier–Stokes system
Findsuch that,
where η^{n+1} = η(c) and ϱ^{ℓ} = ϱ(c^{ℓ}), for ℓ = n and ℓ = n + 1.
B. Space Discretization
For the space discretization, we use the Galerkin method and the finite elements method. Let , , , and be finite elements approximation spaces of , , , and , respectively. As the velocity satisfies homogeneous Dirichlet boundary conditions on Γ, we define the following approximation space:
To simplify the notation, we introduce also the following space:
We require some standard assumptions on approximation spaces:
(2.3)
(2.4)
Moreover, we assume that , and that the L^{2}-projection (resp. ) on the approximation space (resp. ) is stable in H^{1}, that is, there exists a positive constant C independent of h such that:
(2.5)
We assume that the approximation space for order parameters satisfies an inverse inequality: there exists a function C_{inv} of h such that
(2.6)
This property is (for instance) satisfied when the mesh family is quasiuniform, and the approximation spaces are associated to corresponding Lagrange finite elements; in this case, we can choose C_{inv} (h) = C(1 + |ln(h)|) for d = 2 and C_{inv} (h) = Ch^{−1} for d = 3 where C is a constant, which only depends on the mesh regularity (cf. Ref. [18, 4.5.11 (p. 112) and 4.9.2 (p. 123)]).
Finally, we assume that the approximation spaces for velocity and pressure satisfy the so-called uniform inf–sup condition: there exists a positive constant Θ (independent of h) such that
(2.7)
We begin with the definition of discrete functions and at the initial time satisfying:
(2.8)
These discrete functions c and u can be obtained from initial conditions c^{0} and u^{0} by H^{1}(Ω) projection, or as it is the case in practice, by finite elements interpolation provided that c_{i}^{0} and u^{0} are smooth enough.
Assume now that and are given, the Galerkin approximation of Problem 1 reads:
Problem 2 (Formulation with three-order parameters).
Step 1: resolution of the Cahn–Hilliard system
Finds.t. , , fori = 1,2,3,
where α_{jh}is the constant defined by .
Step 2: resolution of the Navier–Stokes equation
Findsuch that , ,
where η = η(c) and ϱ = ϱ(c), for ℓ = nand ℓ = n + 1.
Remark 2. For the resolution of the Cahn–Hilliard system, it is equivalent to only solve the equations satisfied by (c, c, μ, μ) and then to deduce the unknowns (c, μ) using the following relationships:
The resolution of the Navier–Stokes system remains unchanged (cf. Problem 2). In the sequel, in systems where only the unknowns (c, μ, c, μ) are present, the notationcstands for the vector (c, c, 1 − c − c).
III. UNCONDITIONAL STABILITY OF THE SCHEME
We prove in this section the energy equality which ensures the unconditional stability of the scheme.
Proposition 3.1 (Discrete energy equality). Letand . Assume that there exists a solution (c, μ, u, p) of Problem 2. Then, we have the following equality:
(3.1)
whered^{F}(·,·) is the vector (d(·,·))_{i=1,2,3}and
(3.2)
Proof. The key point of the proof is the following observation: the Cahn–Hilliard and Navier–Stokes systems can be rewritten using the function u* defined by (3.2). Then, standard estimations for the Cahn–Hilliard and Navier–Stokes systems are done (Steps 1 and 3) and an estimation on the L ^{2} norm of u* gives the conclusion (Step 2).
Step 1: Owing to the definition (3.2) of the function u*, we observe that the system (2.9) can be rewritten as follows:
We take ν = μ and as test functions in this system. After some standard calculation (see Ref. [9]), this yields:
(3.3)
Step 2: It is possible to obtain an estimation of the first term of the right-hand side of the previous equality. By definition of u*, we have . Multiplying by , and integrating on Ω, yields:
(3.3)
Step 3: The system (2.10) can also be rewritten using the function u*:
We take ν = u and ν = p as test functions in this system. This yields:
(3.5)
The conclusion is obtained by summing up Eqs. (3.3)–(3.5). □
Remark 3. An important difference with the work presented in Ref. [15] in the case of a homogeneous diphasic Cahn–Hilliard/Navier–Stokes model is that no condition is required on the time step to ensure the stability.
The previous stability result enables to prove the existence of solutions for the nonlinear approximate Problem 2.
Theorem 3.2. Given , , we assume that
the coefficients (Σ_{1}, Σ_{2}, Σ_{3}) satisfy (1.4), the mobility satisfies (1.6), and the Cahn–Hilliard potential F satisfies (A),
the discretization of nonlinear termsd^{F}satisfies (2.2) and the following property: there exists (eventually depending on c) such that
(3.6)
Then, there exists at least one solution (c, μ, u, p) to Problem 2.
The proof relies on the following lemma from the topological degree theory [19].
Lemma 3.3 (Topological degree). Let W be a finite dimensional vector space, G be a continuous function fromWtoW. Assume that there exists a continuous function H fromW × [0; 1] to Wsatisfying
i.H (·, 1) = G and H (·, 0) is affine,
ii.∃R > 0 s.t. , if H (w, δ) = 0 then |w|_{W} < R,
iiithe equation H (w, 0) = 0 has a solutionw ∈ W.
Then, there exists at least one solutionw ∈ Wsuch that G(w) = 0 and |w|_{W} < R.
The idea is to link the nonlinear discrete problem to a more simple (linear) problem (using an homotopy, function H of Lemma 3.3) for which we are able to prove existence of solutions (assumption (ii) of Lemma 3.3). The topological degree theory allows to deduce the existence of solutions for the nonlinear problem from a priori estimates, which are in our case deduced from the energy equality (3.1) proved in Proposition 3.1. Such a methodology was used for the approximation of the triphasic Cahn–Hilliard model in Ref. [9]. We only give here the main steps of the proof.
α) Problem 2 is reformulated to enter in the framework of Lemma 3.3. Let W be a finite dimensional vector space . We define a norm on W: for all w = (c_{1h}, c_{2h}, μ_{1h}, μ_{2h}, u_{h}, p_{h}) ∈ W,
and we introduce the function H such that
where and , (resp. and , resp. , resp. ) are defined with their coordinates in the finite elements basis (resp. , resp. , resp. ) of (resp. , resp. , resp. ):
with M = M_{0}((1 − δα)c +δαc), ϱ = ϱ((1 − δ)c +δc) for ℓ = n or ℓ = n + 1 and η = η((1 − δ)c +δ c). The function G is defined by G: w ∈ W → H (w, 1) ∈ W. The problem “Find w^{n+1} such that G (w^{n+1}) = 0” is equivalent (by definition of the function H) to Problem 2. To prove the theorem, we are going to prove that the functions H and G satisfy the assumptions of Lemma 3.3. The continuity of the function H is obtained using the continuity of the different non linear functions (D, ϱ and η) and the Lebesgue's theorem. The function H (ℓ, 0) is clearly affine by construction. β) Let (w^{n+1}, δ) ∈ W × [0; 1] such that H (w^{n+1}, δ) = 0. Note that H (w^{n+1}, δ) = 0 is equivalent to say that w^{n+1} = (c, c, μ, μ, up) is a solution of a problem closely related to Problem 2. The same calculations as in the proof of Proposition 3.1 allows to prove the following estimate:
where . Using Proposition 1.1, the fact that F is nonnegative, the positive lower bounds ϱ_{min} and η_{min} for the density and viscosity, the fact that the mobility is bounded from below, the Korn lemma (cf. Ref. [12, lemma VII.3.5]) and assumption (3.6), we can readily derive the following estimates
(3.7)
where is a constant independent of δ and w^{n +1}. The bound on pressure is obtained using the bound on the velocity (3.7) and the inf–sup condition (2.7), which ensures (cf. Ref. [18, 21.5.10, p. 344]) that there exists such that
Thus, taking ν = v_{h} in the system associated with H(w^{n+1}, δ) = 0 enables to bound the L^{2} norm of the pressure:
(3.8)
where is a constant.
Combining (3.7) and (3.8), we obtain a positive constant independent of δ and c such that
Hence, taking guarantees that for all (w, δ) ∈ W × [0; 1], H(w,δ) = 0⇒|w|_{W} < R.
γ) It remains to prove the existence of a solution to the linear problem H(w^{n+1}, 0) = 0. This problem can be written as three problems, which are totally uncoupled and the existence of solutions for each of these problems is readily obtained (using inf–sup condition).
This concludes the proof of the existence of approximate solutions.
IV. NUMERICAL EXPERIMENTS
In this section, we provide 2D numerical simulations to illustrate the unconditional stability stated in Proposition 3.1.
The space discretization is performed on square local adaptive refined meshes using:
Lagrange finite element for the order parameters c_{1}, c_{2}, c_{3}, the chemical potentials μ_{1}, μ_{2}, μ_{3} and for the pressure p,
Lagrange finite elements for each component of the velocity u.
The adaptation procedures are based on conforming multilevel finite element approximation spaces that are built by refinement or unrefinement of the finite element basis functions instead of cells. All the details about this method and also various examples (in particular, simulations using the Cahn–Hilliard model considered in this article) are described in Ref. [20]. The refinement criterion used in those (un-)refinement procedures imposes the value of the smaller diameter h_{min} of a cell and ensures that refined areas are located in the neighborhood of the interfaces (i.e., where no order parameter is equal to one). We do not give more details on spatial discretization issues here, as the main goal of this article is to investigate the properties of time discretization schemes.
We compare the results obtained with the unconditionally stable scheme (denoted Uncond. in the sequel) proposed in this article (see Problem 2) and the scheme (denoted Stand. in the sequel) used in Refs. [11, 15] which is obtained using an explicit velocity in the Cahn–Hilliard equation (see “Coupling Terms” section).
The (nonlinear) Cahn–Hilliard system is solved using the Newton algorithm and the Navier–Stokes system is solved using the Augmented-Lagrangian method. All intermediates linear system are solved using direct solvers.
The practical implementation has been performed using the software object-oriented component library PELICANS [21], developed at the “Institut de Radioprotection et de Sûreté Nucléaire (IRSN)” and distributed under the CeCILL-C license agreement (an adaptation of LGPL to the French law).
A. Droplet Oscillation
The first example is a two-phase flow simulation of the oscillations of a 2D droplet due to surface tension. This test case was already used in several articles, see for instance Refs. [22–25]. The initial configuration is a 2D droplet with a perturbed radius: r = r_{0}(1 +α cos(2θ)) (in polar coordinates). More precisely, we choose the square] − 4r_{0},4r_{0}[^{2} as computational domain and initialize the order parameters c_{1} and c_{2} with the following formula:
where the interface width ε is given by . We use the values r_{0} = 0.1 and α = 0.05 in all our simulations.
The perturbed droplet is initially at rest and the only external force is the surface tension σ = 1 (i.e., there is no gravity g = 0). We assume that the two phases have the same densities ϱ_{1} = ϱ_{2} = 1 and the same small viscosities η_{1} = η_{2} = 10^{−4}. We take a small constant mobility M_{0} = 10^{−5} and perform simulations until the final time T = 0.2. The space discretization is fixed: .
Figure 1 shows the time evolution of the kinetic energy (on the left) and of the total energy (on the right) using the Uncond. scheme for different values of the time step Δt.
The Uncond. scheme ensures the decrease of the total energy for all time steps. This is not the case when using Stand. scheme. Figure 2 shows a comparison of the energies evolution between Uncond. scheme and Stand. scheme. For Δt = 10^{−4}, the results are very similar but for Δt = 2 × 10^{−4} or greater, the Stand. scheme leads to a blow up of kinetic and total energies.
Figure 3 shows the interface shape and the streamlines (of velocity) at t = 0.04 that we obtain when using the Uncond. scheme (on the left) and the Stand. scheme (on the right). These pictures show 20 contour levels of the order parameter c_{1} between 0.4 and 0.6, and 50 contour levels of the streamline function. It appears small instabilities in the neighborhood of the interface when using the Stand. scheme.
B. Two-Dimensional Three-Phase Flow
The second example is a three-phase flows simulation of a gas bubble rising in a liquid column under gravity. Physical properties of the three phases and initial configuration are given in Fig. 4. The interface width is given by and we choose a degenerate mobility, that is, the mobility vanishes in pure phases: M_{0}(c) = 10^{−5}(1 − c_{1})^{2}(1 − c_{2})^{2}(1 − c_{3})^{2}. We take Δt = 5 × 10^{−5} and for this simulation.
The bubble rises in the heavy liquid (phase ****), penetrates the liquid–liquid interface and then rises in the light liquid (phase ***) entraining some quantity of the heavy liquid in the upper phase. This time evolution is shown in Fig. 5 where simulations performed with the Uncond. scheme (on the left) and the Stand. scheme (on the right) are compared. These pictures show 50 contour levels of the order parameters c_{1} (in red) and c_{2} (in blue) between 0.4 and 0.6 (be careful the contour levels of c_{1} and c_{2} may coincide).
The results we obtain with the two schemes are close: the bubble rises with the same velocity in the two cases. However, we observe two main differences: the first is the form of the bubble (see for instance picture at t = 0.6), and the second is the time at which the column of entrained fluid break up (see pictures at time t = 0.75 and 0.9). These differences are certainly due to the fact that the stabilization modifies the value of the mobility in the Cahn–Hilliard equation and consequently the Uncond. scheme involves a more important diffusion at the interface than the Stand. scheme. Nevertheless, note that in the right picture at t = 0.9, the thickness of the entrained liquid column is equal to the mesh size, so the column is near to break up.
V. CONVERGENCE OF APPROXIMATE SOLUTIONS IN THE HOMOGENEOUS CASE
In this section, we assume that ϱ_{1} = ϱ_{2} = ϱ_{3} = ϱ_{0} > 0. This implies that the function ϱ(c) is constant: .
The existence of solutions is given by Theorem 3.2. For all , we can introduce the following piecewise constant and piecewise linear interpolations in time:
Note that, in the sequel, c, μ, and u always designate the functions of the time variable t defined above, which are naturally indexed with the size of the time discretization. To avoid confusion, we designate the particular values of c, μ, and u for n = N with the notation c, μ, and u. The convergence result is the following:
Theorem 5.1 (Convergence theorem). We assume that assumptions of Theorem 3.2 are satisfied, so that a solution (c, μ, u, p) of Problem 2 exists for alland for allh > 0. We assume that , that the consistency property (2.1) is satisfied and that there exist two constantsC > 0 and Δt_{0} > 0 such that for all Δt ≤ Δt_{0}and for alln ∈ [[0; N − 1]],
(5.1)
Consider the problem (1.8), the initial conditions (1.11) and the boundary conditions (1.9)-(1.10). Then, for allt_{f} > 0, there exists a weak solution (c, μ, u, p) defined on [0, t_{f}[ such that
Moreover, for all sequencesandsatisfying the following properties:
and ,
there exists a constantA (indep. of K) s.t.: (recall that C_{inv}is given by (2.6))
(5.2)
the sequences , andsatisfy, up to a subsequence, the following convergences whenK→ + ∞:
Remark 4. The assumption (5.1) is obtained in practice by applying Proposition 3.1 and by bounding the term:
in the right-hand side of (3.1). The way to obtain this bound depends on the schemeD(c^{n}, c^{n+1}) chosen for the nonlinear terms of the Cahn–Hilliard system. This was largely discussed in Ref. [9].
Remark 5. In the statement of Theorem 5.1, the inequality (5.2) is not a stability condition. When using a quasiuniform mesh family and the associated Lagrange finite elements in 2D, this condition is not restrictive, because we can choose C_{inv} (h) = C(1 + |ln(h)|). In 3D, it means that to obtain convergence toward weak solutions of the continuous problem, it is necessary that the time step goes to zero faster than the mesh size.
The proof of Theorem 5.1 is inspired from Refs. [14, 15] which deal with the homogeneous diphasic Cahn–Hilliard/Navier–Stokes system. Excluding the fact that we are interesting in a triphasic model, the major difference with these works is the taking into account of the transport term in the Cahn–Hilliard equation. We have to prove that the additional term do not disturb the consistency.
Basically, the proof of Theorem 5.1 is split in three steps:
first, the energy equality (5.1) allows to prove that the sequences , , and are bounded in some suitable norms.
it is then possible to apply compactness theorems to extract some convergent subsequences.
the third step consists in proving that the obtained limit is a weak solution of the system (1.8).
We give the details for each of these three steps below. In the sequel (subsections 5.1–5.3 in Sections V), we assume that assumptions of Theorem 5.1 are satisfied and in particular the notation c, μ, u, p… denote solutions of the discrete Problem 2, and , , the associated sequences.
A. Bounds on Discrete Solution
In this section, we assume that K is fixed and to simplify notation we omit the index K in the notation h_{K} and N_{K}. The first estimates stated in Proposition 5.2 are directly derived from the energy estimate (5.1).
Proposition 5.2. We have the following bounds:
(5.3)
(5.4)
(5.5)
whereK_{1}, K_{2}, andK_{3}are three constants independent of Δtandh.
Proof. This proof is very similar to the one of Proposition 4.2 in Ref. [9]. Nevertheless, note that it use additional ingredients (Korn lemma (cf. Ref. [12, lemma VII.3.5]), the lower bound for the viscosity η(c) and the fact that the density is constant) to deal with the terms that involve the velocity. □
To pass to the limit in nonlinear equations (cf. subsection 5.3 in Section V), we need strong convergence of the subsequences. For this reason, it is useful to obtain more accurate estimates.
Proposition 5.3. There exist two constantsK_{4}andK_{5}independent ofhand Δtsuch that:
(5.6)
(5.7)
Proof.
i.The estimate (5.6) is obtained from the first equation of the Cahn–Hilliard system.(α) Consider . The first equation of (2.9) reads: Thus, the inverse inequality (2.6) yields: Finally, thanks to (5.2) and (5.3), we obtained that there exists a constant K (independent of h and Δt) such that:
(5.8)
We are now going to use this intermediate inequality to prove (5.6).(β) Let ν ∈ H^{1}(Ω). Let ν be the L^{2} projection of ν on . Owing to (2.5), we have: Thus, using (5.8), we obtain As this inequality is true for all ν ∈ H^{1}(Ω), we have Consequently, using (5.4) yields:
(5.9)
(γ) We now take ν = Δt(c − c) in (5.8). This yields: and so, using (5.4) and (5.5), we have:
(5.10)
The inequality (5.6) is readily deduced from Eqs. (5.9) and (5.10) by defining the constant .
ii.To obtain estimate (5.7), we begin with bounding the term: for n ∈ [[0;N − i − 1]]. We choose such that
(5.11)
as a test function in (2.10) and we sum up the equations to obtain:
We then separately estimate each term of this inequality. For term T_{1}, by using the Hölder inequality and an interpolation inequality, we obtain:
Using the bound (5.3) and the Young inequality yields:
We conclude using the Hölder inequality and the bound (5.4):
The term T_{2} is bounded in the same way:
For the viscous term T_{3}, we derive the following estimate:
It remains the terms T_{4} and T_{5}:
and
Finally, we obtain the following result: there exists a positive constant K such that, for all satisfying (5.11), we have
In particular, for ν = u − u (which satisfies (5.11) owing to (2.10)), we find
Thus, we obtain
This leads to the conclusion with . □
B. Compactness Argument, Convergence of Subsequences
The estimates proved in Subsection A in Section V (Propositions 5.2 and 5.3), allow to obtain (up to subsequences) the convergence of the sequences: , , , , , , and . The following propositions give the spaces in which these convergences hold.
Proposition 5.4. Up to subsequences, we have the following convergences whenK → +∞:
(5.12)
(5.13)
(5.14)
(5.15)
Proof. The convergences (5.12)–(5.15) are direct consequences of Proposition 5.2. Indeed, it is easy to verify that the estimates stated in this proposition prove that the sequences , , , and are, respectively, bounded in the following norm: L^{∞}(0, t_{f}, (H^{1}(Ω))^{3}), L^{2}(0, t_{f}, (H^{1}(Ω))^{3}), L^{2}(0, t_{f}, (H^{1}(Ω))′), and L^{2}(0, t_{f}, (H^{1}(Ω))^{d}).□
The weak convergences we write above are not sufficient to pass to the limit in the nonlinear terms of the Cahn–Hilliard and Navier–Stokes systems. We prove in the next two propositions that it is possible to obtain strong convergence for order parameters and velocity in some suitable function spaces.
Proposition 5.5. Up to subsequences, we have the following convergences whenK → +∞:
(5.16)
(5.17)
Proof. The sequence is bounded in L^{∞}(0, t_{f}, (H^{1}(Ω))^{3}) and its time derivative is bounded in L^{2}(0, t_{f}, (H^{1}(Ω))′). We obtain the strong convergence (5.16) of order parameters by applying the Aubin–Lions–Simon compactness theorem [26]. From this convergence and using the inequality (5.5), we deduce the strong convergences (5.17). □
To prove the result of strong convergence on the velocity, we need to apply a more precise compactness result, because we do not have any estimate on its time derivative. We apply a compactness theorem due to Simon in which the condition on the time derivative is replaced by an estimation on time translates.
First, we write the term to estimate. This term is defined from the discrete function u which is piecewise constant (in time) and its time translate. We link this term to the values u of the function on each time intervals to exploit estimates proved in Subsection A in Section V. To simplify the notation, we omit in this lemma, the index K in the notation h_{K} and N_{K}.
Lemma 5.6. Let τ ∈]0, t_{f}[. We denote byi ∈ [[0; N − 1]] the unique index suchthat t^{i} ≤ τ < t^{i+1}. Then, we have:
iif τ < Δtthen
iiin all cases, we have:
We can now state the proposition giving the strong convergence for the velocity.
Proposition 5.7. Up to subsequences, we have the following convergences whenK → +∞:
(5.18)
Proof. The proof relies on a compactness theorem due to Simon [26, Theorem 5, p.84] which state that the embedding
is compact. The Nikolskii space is defined by:
with the norm
Thus, since the sequence is bounded in the spaces L^{2}(]0, t_{f}[,(H^{1}(Ω))^{d}) and L^{2}(]0, t_{f}[,(L^{2}(Ω))^{d}) (cf. Eqs. (5.3) and (5.4)), it is sufficient to prove that it is bounded in the space , to obtain the conclusion. Let τ ∈]0, t_{f}[. We still omit the index K in the notation h_{K} and N_{K}.
i.If τ < Δt then owing to Lemma 5.6, we have:
ii.If τ ≥ Δt then owing to Lemma 5.6, and then using the inequality (5.7), we have: since we have t^{i} ≤ τ and t^{i+1} = t^{i} + Δt ≤ 2τ.
In all cases, we obtained the existence of a positive constant K_{6} (independent of h and Δt) such that:
This concludes the proof of the convergence of u. The convergences of and are then obtained thanks to the inequality (5.5). □
C. Passing to the Limit in the Scheme
The convergences obtained in Subsection B in Section V allow to pass to the limit in the discrete system.
For the Cahn–Hilliard system (without the transport term), this work was already done in details in Ref. [9]. We focus here on the transport term and on the Navier–Stokes equation.
To simplify the notation, we still omit the index K in the notation N_{K} and h_{K} but when we say “convergence” it means K → +∞ (and consequently N_{K} → +∞ and h_{K} → 0).
Transport Term in the Cahn–Hilliard Equation.
Let a given function and . We define ν as the H^{1} projection of the function ν^{μ} on . We have to prove the following convergence:
(5.19)
We proceed in two steps, separately considering two terms of the left-hand side: the standard transport term and the additional term that ensures the unconditional stability.
The following inequalities allows to identify the limit of the first term:
as c is bounded in L^{2}(0, t_{f}, H^{1}(Ω)), u is bounded in L^{2}(0, t_{f}, (H^{1}(Ω))^{d}), c (strongly) converges in L^{2}(0, t_{f}, L^{2}(Ω)) toward c_{i} (cf. Eq. (5.17)), u (strongly) converges in L^{2}(0, t_{f}, (L^{2}(Ω))^{d}) toward u (cf. Eq. (5.18)) and, owing to assumption (2.3), .
We now use the fact than the sequences c are μ are, respectively, bounded in L^{∞}(0, t_{f}, H^{1}(Ω)) and L^{2}(0, t_{f}, H^{1}(Ω)) norm, the inverse inequality (2.6) and the condition (5.2) on the sequences h_{K} and N_{K} to show that the second term convergences toward 0:
Thus, we proved that the convergence (5.19) holds. Reusing (exactly as it is) the reasoning presented in Ref. [9] allows to pass to the limit in the other terms of the Cahn–Hilliard system.
Navier–Stokes System.
Let satisfying div (ν^{u}) = 0 and such that τ(t_{f}) = 0.
We introduce the space
The inf–sup condition (2.7) implies that the function ν^{u} ∈H(Ω) which is divergence free can be “well approximated” with functions in Z_{h}. This is detailed in Proposition 5.8.
Proposition 5.8 (Approximation of divergence free functions, [18, eq. 12.5.17]). We have the following inequality:
Let ν be the H^{1} projection of ν^{u} on the space Z_{h}. Proposition 5.8 and the assumption (2.4) show that
(5.20)
We use ν as a test function in the first equation of (2.10). We then multiply by τ(t), t ∈]t^{n}, t^{n+1}[, integrate between t^{n} and t^{n+1}, and sum up for n from 0 to N − 1, so that we rebuilt a variational formulation on]0, t_{f}[ × Ω. The unsteady term is modified by a discrete integration by part:
Thus, we obtain the following formulation of the scheme in which we can pass to the limit:
The limit of the term T_{1} is readily obtained from strong convergences (5.18), (5.20) and those of functions toward τ′ in L^{2}(0, t_{f}) (obtained for instance with dominated convergence theorem since the function τ is in ):
The term T_{2} allows to show that u satisfies the initial condition (1.11) in a weak sense. The convergences (2.8), (5.20) and the uniform convergence on [0, t_{f}] of the function t ↦ τ(Δt(t − 1)) toward the constant function equal to τ(0) yields:
Concerning the term T_{3}, the following inequality allows to conclude:
Indeed, as the sequences (u) and are bounded in L^{2}(0, t_{f}, (H^{1}(Ω))^{d}), the convergences (5.18) and (5.20) show that the first two terms of the above right-hand side tend to 0. The last one (the term involving the integral) also tends to 0 by weak convergence of toward ∇u in L^{2}(0, t_{f}, (L^{2}(Ω))^{d}) (a component-by-component reasoning gives the result, as for all 1 ≤ i, j ≤ d, the function (t, x) ↦ u_{i}(x)ν(x)τ(t) lies in L^{2}(0, t_{f}, L^{2}(Ω))).
The term T_{4} is treated in the same way:
the conclusion being now obtained using the convergences (5.18), (5.20) and the fact that the two sequences u and are bounded in L^{2}(0, t_{f}, (H^{1}(Ω))^{d}).
The limit of the term T_{5} is obtained using the following convergence (up to a subsequence):
(5.21)
This convergence is proved using the dominated convergence theorem (the viscosity η is a bounded continuous function and strongly converge in L^{2}(0, t_{f}, (L^{2}(Ω))^{3}), almost everywhere up to a subsequence).
Thus, using (5.20), (5.21), the fact that the sequence is bounded in L^{2}(0, t_{f}, (H^{1}(Ω))^{d}), and the weak convergence of toward Du in L^{2}(0, t_{f}, (L^{2}(Ω))^{d}), we obtain
By (5.20), the convergence of term T_{6} is straightforward:
The convergence of the capillary forces term T_{7} is obtained as follows:
The first two terms of the right-hand side tend to 0 thanks to convergences (5.17) and (5.20), as the sequences (c) and (μ) are bounded in L^{2}(0, t_{f}, H^{1}(Ω)) and L^{∞}(0, t_{f}, H^{1}(Ω)), respectively. The last term tends to 0 by weak convergence of ∇μ towards ∇μ_{j} in L^{2}(0, t_{f}, (L^{2}(Ω))^{d}).
Finally, it only remains to prove that the residual term T_{8} tends to 0. This simply comes from the fact that:
and
In conclusion, we proved that:
To finish, passing to the limit in the constraint equation yields:
VI. CONCLUSIONS
We proposed in this article an original scheme for the discretization of a triphasic Cahn–Hilliard/Navier–Stokes model.
This scheme is unconditionally stable and preserves, at the discrete level, the main properties of the model, that is the volume conservation and the fact that the sum of the three-order parameters remains equal to 1 during the time evolution.
We proved the existence of at least one solution of the discrete problem and, in the homogeneous case (i.e., three phases with the same densities), we proved the convergence of discrete solutions toward a weak solution of the model (whose existence is proved in the same time).
The main perspective is the study of the convergence in the case where the three fluids in presence have different densities. Even if the energy estimate (and the existence of discrete solutions) are still true in this case, it is delicate to obtain sufficient estimates which would lead, by compactness, to strong convergence on the velocity which is necessary to pass to the limit in nonlinear terms. Indeed, the Navier–Stokes equation involves a term of the form u∂_{t}ϱ. The time derivative of the density is not very smooth, because it is a function of order parameters whose time derivative is in L^{2}(0, t_{f}, (H^{1}(Ω))′).
Acknowledgements
The author expresses his gratitude to Franck Boyer and Bruno Piar for their support during the preparation of this work and thanks the referees for their careful reading of the article.